We turn now to a more realistic example of the structures within an expanding cosmological background. To model such a structure, let us begin with a perturbed FRW model
representing the dynamics of a structure within the expansing universe in the linear regime[1]}. We use the so-called Newtonian gauge and write the perturbed FRW metric keeping first order terms in the metric functions and velocities (\(\overline{u_r^2},\overline{u_\phi ^2},\overline{u_\theta ^2} \ll \overline{u_t^2} \approx 1\) ):
\(&ds^2=-(1+2\Phi )~dt^2+a(t)^2(1-2\Phi )~(dr^2+r^2d\Omega ^2).\\\) 
